WASHINGTON—Senate Majority Leader Mitch McConnell’s insistence that the next president fill the sudden Supreme Court vacancy has won broad support from fellow Republicans, opening a new front in election-year battles with Democrats where control of the Senate is at stake.

The fight over the opening created by Justice Antonin Scalia’s death already has moved to the center of the presidential campaign. But it now also figures to be just as prominent in key Senate races, particularly for vulnerable Republican incumbents in states...

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